scholarly journals A Class of Unbounded Fourier Multipliers on the Unit Complex Ball

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Pengtao Li ◽  
Jianhao Lv ◽  
Tao Qian
Keyword(s):  
Unit Sphere ◽  
Fourier Multiplier ◽  
Differential Operators ◽  
Fourier Multipliers ◽  
Complex Ball ◽  
Fractional Differential ◽  
Fourier Multiplier Operators ◽  
Fractional Differential Operators ◽  
Multiplier Operators

We introduce a class of Fourier multiplier operatorsMbonn-complex unit sphere, where the symbolb∈Hs(Sω). We obtained the Sobolev boundedness ofMb. Our result implies that the operatorsMbtake a role of fractional differential operators on∂𝔹.

scholarly journals Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Ravi P. Agarwal ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Series Solution ◽  
Differential Operators ◽  
Fractional Wave Equation ◽  
Local Fractional Calculus ◽  
Vibrating String ◽  
Fractional Differential ◽  
Fractional Differential Operators

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

scholarly journals Weighted Morrey Estimates for Multilinear Fourier Multiplier Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Songbai Wang ◽  
Yinsheng Jiang ◽  
Peng Li
Keyword(s):  
Fourier Multiplier ◽  
Morrey Spaces ◽  
Fourier Multipliers ◽  
Sobolev Regularity ◽  
Morrey Estimates ◽  
Fourier Multiplier Operators ◽  
Multiplier Operators

The multilinear Fourier multipliers and their commutators with Sobolev regularity are studied. The purpose of this paper is to establish that these operators are bounded on certain product Morrey spacesLp,k(ℝn). Based on the boundedness of these operators fromLp1(ω1)×⋯×Lpm(ωm)toLp(∏j=1m‍ωp/pj), we obtained that they are also bounded fromLp1,k(ω1)×⋯×Lpm,k(ωm)toLp,k(∏j=1m‍ωp/pj), with0<k<1,1<pj<∞,1/p=1/p1+⋯+1/pm, andωj∈Apj,j=1,…,m.

scholarly journals Spectral Approximation of Fractional PDEs in Image Processing and Phase Field Modeling

2017 ◽  
Vol 17 (4) ◽  
pp. 661-678 ◽  
Author(s):  
Harbir Antil ◽  
Sören Bartels
Keyword(s):  
Image Processing ◽  
Phase Field ◽  
Differential Operators ◽  
Mathematical Tool ◽  
Phase Field Models ◽  
Field Modeling ◽  
Regularity Properties ◽  
Fractional Pdes ◽  
Fractional Differential ◽  
Fractional Differential Operators

AbstractFractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. The numerical solution of corresponding model problems via a spectral method is analyzed. Its efficiency and features of the model problems are illustrated by numerical experiments.

scholarly journals The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012031
Author(s):  
E.A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Differential Operators ◽  
Space Time ◽  
Partial Differential ◽  
Stochastic Problems ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Time Fractional Differential Equations

Abstract The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

Dynamical behavior of fractionalized simply supported beam: An application of fractional operators to Bernoulli-Euler theory

2021 ◽  
Vol 10 (1) ◽  
pp. 231-239
Author(s):  
Kashif Ali Abro ◽  
Abdon Atangana ◽  
Ali Raza Khoso
Keyword(s):  
Differential Operators ◽  
Special Functions ◽  
Dynamical Behavior ◽  
Analytic Solutions ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Parametric Resonances ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Graphical Illustration

Abstract The complex structures usually depend upon unconstrained and constrained simply supported beams because the passive damping is applied to control vibrations or dissipate acoustic energies involved in aerospace and automotive industries. This manuscript aims to present an analytic study of a simply supported beam based on the modern fractional approaches namely Caputo-Fabrizio and Atanagna-Baleanu fractional differential operators. The governing equation of motion is fractionalized for knowing the vivid effects of principal parametric resonances. The powerful techniques of Laplace and Fourier sine transforms are invoked for investigating the exact solutions with fractional and non-fractional approaches. The analytic solutions are presented in terms of elementary as well as special functions and depicted for graphical illustration based on embedded parameters. Finally, effects of the amplitude of vibrations and the natural frequency are discussed based on the sensitivities of dynamic characteristics of simply supported beam.

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